Bill martin worcester polytechnic institute usa
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Bill Martin Worcester Polytechnic Institute USA. Cometric Association Schemes. Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008. Several Collaborators. Jason Williford Misha Muzychuk Edwin van Dam Nick LeCompte (WPI student) Will Owens (WPI student)

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Cometric Association Schemes

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Bill Martin

Worcester Polytechnic Institute

USA

Cometric Association Schemes

Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008


Several Collaborators

  • Jason Williford

  • Misha Muzychuk

  • Edwin van Dam

  • Nick LeCompte (WPI student)

  • Will Owens (WPI student)

  • . . . and I’ve received valuable suggestions from many others.


Today’s Goals

  • Survey the known examples

  • Summarize the main results to date

  • Explore the structure of imprimitive

    Q-polynomial schemes, especially with

    3 or 4 classes

  • List some open problems, big and small


My Real Goals

  • To make the next 45 minutes as pleasant as possible


My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)


My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb


My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb

  • To get some smart people to work on these interesting problems


My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb

  • To get some smart people to work on these interesting problems

  • To tell you as much as I reasonably can about the subject


My Real Goals

  • To make the next 45 minutes as pleasant as possible (for both you and me)

  • To not look too dumb

  • To get some smart people to work on these interesting problems

  • To tell you as much as I reasonably can about the subject

  • To avoid typesetting math in PowerPoint


First, an Example

E8 Root Lattice


A Spherical 7-Design in R8


A Curious Property


Krein Parameters in Disguise


The Polytope Definition


The Polytope Definition


The Polytope Definition

Inner product of two zonal polynomials only depends on distance between the two base points and the single-variable polynomials.


Another Example


The Usual Approach


The Bose-Mesner Algebra


Orthogonality Relations


A Taste of Duality


Polynomial Schemes

Delsarte (1973):


Some Natural Questions

Concerning cometric association schemes . . .


What do they look like?

  • I don’t know

  • The model I just showed you is my favorite definition so far


Balanced Set Condition


Balanced Set Condition

Terwilliger (1987):


Sources of Examples

  • Q-polynomial distance-regular graphs (e.g., all those with classical parameters)

  • Spherical designs / lattices

  • Extremal codes and block designs

  • Real mutually unbiased bases

  • Sporadic groups (e.g., triality)

  • linked systems of designs and geometries


How Many?


Are They Mostly P-Polynomial?


Cometric Schemes with Large d


Cometric Schemes with Large d


What do the Imprimitive Cometric Schemes Look Like?


Duality and Imprimitivity

w=3 fibres of size r=2

w=2 fibres of size r=3

A familiar dual pair of association schemes


Duality and Imprimitivity

Another dual pair of complete multipartite schemes


Suzuki’s Theorem

H. Suzuki (1998):


Q-Bipartite Structure


Diam. 3 Distance-Regular Graphs


3-Class Cometric Schemes


3-Class Cometric Schemes

Edwin van Dam (1995)


Diam. 4 Distance-Regular Graphs


4-Class Cometric Schemes


Don Higman’sTriality Scheme


Donald Higman (1928-2006)


More 4-Class Q-Antipodal Schemes


Hyperovals in PG(2,4)

This is a 4-class Q-antipodal association scheme


Schemes from Unbiased Bases


Real MUBs


Real MUBs


A Surprising Result


Four-Class Schemes from MOLS

A Construction of Wocjan and Beth (2005)


Four-Class Schemes from MOLS

A Construction of Wocjan and Beth (2005)


Four-Class Schemes from MOLS


Four-Class Schemes from MOLS


Thomas Beth, 1949−2005


Linked System of Symmetric Designs

  • 48 vertices, split into three classes of size 16

  • Graph G1represents “incidence”, yielding a

    square (16,6,2)-design between any two

    Q-antipodal classes

  • “linked”: the number of common neighbors in the third class of a point chosen from Class One and a point chosen from Class Two depends on only whether or not these are incident (1 and 3, resp.)


MUBs from Cameron-Seidel Scheme

  • Muzychuk, Williford, WJM introduced the extended Q-bipartite double

  • Applied to the subschemes of the Cameron-Seidel scheme, these are 4-class Q-bipartite, Q-antipodal schemes

  • So we have the same schemes that Bannai and Bannai found from mutually unbiased bases


More Material in Beamer Format

Check time available


Thank You!


Shortest Vectors in Leech Lattice

  • 196560 vectors in R24, all of squared length 8

  • only 7 possible inner products: ±8, ±4, ±2, 0

  • construct one graph for each inner product

  • we obtain a 7-class cometric scheme which is Q-bipartite

  • Krein array:

  • {24, 23, 288/13, 150/7, 104/5, 81/4;  

  • 1, 24/13, 18/7, 16/5, 15/4, 24 }


One More Photo

Heather Lewis and …?


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